Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#5912 |
| Fitting | C24×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C24×C4 | D3 |
| socle | C24 | C22×D3 |
| hypercenter | C22×C4 | S4 |
| hypocenter | A4 | C2×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 | |
895 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 895 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2)(3,4,6)(7,9)(8,11)(10,13)(12,14);(7,8,10,12)(9,11,13,14);(4,5)(9,13)(11,14);(1,2)56 conjugacy classes — view the full character table.
| maximal subgroup of | #11712705, #11712706, C2×#73117, … |
| maximal normal subgroup of | #11712705, #11712706, C2×#73117, … |
| maximal quotient of | C2×#85482, C2×#85287, C2×C4×GL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20050);